(CNN) -- Hillary Clinton's lead in the race for the Democratic presidential nomination is shrinking in New Hampshire.

Most observers say Sen. Hillary Clinton did well in last week's debate against Sen. Barack Obama and others.

The senator from New York saw a 23-point lead over her closest rival in September decrease to 14 points in a new CNN/WMUR New Hampshire presidential primary poll conducted by the University of New Hampshire.

When asked who they would support if the primary were held today, 36 percent of likely Democratic primary voters backed Clinton, 22 percent supported Sen. Barack Obama of Illinois and 13 percent favored former Sen. John Edwards of North Carolina.

In September, when CNN last polled in New Hampshire, Clinton was at 43 percent, with Obama at 20 percent and Edwards at 12 percent.

The survey, released Tuesday, showed a jump for Bill Richardson -- 12 percent of those questioned supported the New Mexico governor for president, double his support from September.

The CNN/WMUR poll comes out a day after a poll in Iowa found Obama first in the state, although his lead is within the sampling error.

One reason for Clinton's shrinking lead may be voter perceptions of her as a candidate. Thirteen percent of those polled in New Hampshire said Clinton is the candidate who is most honest and trustworthy, 27 percent said Obama is.

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Clinton also polled poorly in the Iowa survey when it came to who's the most honest and trustworthy -- 15 percent said Clinton was, putting her behind Obama and Edwards.

"I think the race is getting closer for two reasons," said Democratic pollster Geoff Garin. "One is that Barack Obama is doing a better job at making his case and he's got a chance to do it in two states where politics are up close and personal. Secondly, Hillary Clinton went through a period that focused on the thing that voters worry the most about in terms of her, which is does she play it straight?

"So, we've had a period that has really accentuated Sen. Obama's strengths and put some focus on Sen. Clinton's one big area of vulnerability."

Clinton's strong suit is health care. More than half of those polled in the CNN poll said Clinton would handle that issue best.

Twenty-seven percent of those surveyed said Clinton is the best candidate to handle the war in Iraq, the issue that New Hampshire Democrats said is the most important issue facing the country right now. Forty-one percent said the war is the most important issue, with health care second at 23 percent and the economy third at 10 percent.

On the bright side for Clinton, more than two-thirds of those questioned said Clinton, the front-runner in national polls, is the candidate most likely to win the Democratic nomination. And half of those surveyed say the senator from New York is the toughest among the Democratic White House hopefuls in standing up to Republicans.

The poll was conducted between Wednesday and Sunday. Pollsters surveyed 389 New Hampshire residents who said they'll vote in the state's 2008 Democratic primary.

The poll has a sampling error of plus or minus 5 percentage points.

In Iowa, 30 percent of those questioned in the ABC/Washington Post poll supported Obama. Clinton was at 26 percent, Edwards at 22 percent and Richardson at 11 percent.

The poll also found Obama 4 points ahead of Clinton among likely Democratic caucus-goers on the question of which candidate would do a better job handling the war in Iraq.

The margin of error in the survey was 4 percent.

"One issue that appears to hurt Clinton in Iowa is Iraq," said CNN polling director Keating Holland. "Caucus-goers think Obama can handle that issue better than Clinton. The reverse is true in New Hampshire -- although not by much. Twenty-seven percent said Clinton would best handle the war; 21 percent favored Obama.

"That's important," Holland added, "because Iraq is the No. 1 problem in the minds of Democratic voters in both Iowa and New Hampshire."

The Iowa caucuses, which will be held January 3, kick off the presidential primary season. New Hampshire traditionally goes second. The primary is expected to take place January 8, although the date is not set.

While Clinton, Obama and Edwards have been in a dogfight in Iowa for some time, the tighter race in New Hampshire is a recent development.

"New Hampshire has always been Clinton's insurance policy. Many observers believe she can survive a loss in Iowa -- but only if she can finish first in the Granite State," Holland said.E-mail to a friend